# Feynman diagrams for Yukawa Interaction

I'm solving some QFT problems and one of them deals with the Yukawa coupling. It asks to consider three processes, namely, $\psi\psi\to\psi\psi$, $\psi\bar{\psi}\to\psi\bar{\psi}$ and $\phi\phi\to\phi\phi$, and then find the Feynman diagrams to second order and derive the matrix elements.

The issue is that if I understood the coupling, there is one single vertex type and it requires a fermion particle, a fermion anti-particle and a scalar particle to meet at the vertex.

For the first and last processes I simply can't imagine a diagram with the given external legs, given number of internal points (namely one and two) and with the internal pointa subject to the constraint that a $\psi$ line a $\bar{\psi}$ line and a $\phi$ line must meet there. Not even the simple tree level diagrams for the $s,t,u$ channels seems to work.

How can I find the Feynman diagrams for the given processes in Yukawa theory? What is the correct reasoning and the correct diagrams?

I'm assuming you're using the lagrangian

$$\mathcal{L}=\overline{\psi}(i\gamma^{\mu}\partial_{\mu}-m_{\psi})\psi+\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m_{\phi}-g\overline{\psi}\psi\phi,$$

or something similar.

The diagrams don't require you to have a particle-antiparticle-scalar triple to make a vertex. It just requires that all interaction vertices must have two $\psi$ propagators and one $\phi$ propagator. So long as the charge flow into each vertex is valid, you are fine. For instance, think of the first process,

$$\psi\psi\to\psi\psi$$

as a $t$ channel process. It's just the

$$\psi\overline{\psi}\to\psi\overline{\psi}$$

diagram, but rotated on its side (that is, related by a crossing symmetry).

The $\phi\phi\to\phi\phi$ process is actually a bit more complicated, and there isn't a diagram to second order that contributes to it. The lowest order diagram would be a box diagram (see below, with curvy lines representing $\phi$ propagators and straight lines representig $\psi$ propagators).

I hope this helped!

• Thanks, with this I've found for particle-particle scatering the $t,u$ diagrams with a relative sign between them. The $s$ diagram isn't there by charge conservation if I'm right. I didn't find any diagram with loops though, just tree level ones. There should be a diagram with a loop? Or the other diagram with loop will be the $\phi\phi\to\phi\phi$ process as you've outlined? – user1620696 Jun 18 '17 at 21:29